# How to visualize the tasks for the rope and nails puzzle?

Following the A&Q I want to visualize the tasks for the rope and nails puzzle. In the paper, one can see 11 puzzles:

Puzzle 1 (1-out-of-3) Hang a picture on three nails so that removing any one nail fells the picture.

Puzzle 2 (2-out-of-3) Hang a picture on three nails so that removing any two nails fells the picture, but removing any one nail leaves the picture hanging.

Puzzle 3 (1+2-out-of-3) Hang a picture on three nails so that removing the first nail fells the picture, as does removing both the second and third nails, but removing just the second or just the third nail leaves the picture hanging.

Puzzle 4 (1-out-of-4) Hang a picture on four nails so that removing any one nail fells the picture.

Puzzle 5 (2-out-of-4) Hang a picture on four nails so that removing any two nails fells the picture, but removing any one nail leaves the picture hanging.

Puzzle 6 (3-out-of-4) Hang a picture on four nails so that removing any three nails fells the picture, but removing just one or two nails leaves the picture hanging.

Puzzle 7 (2+2-out-of-2+2) Hang a picture on two red nails and two blue nails so that removing both red nails fells the picture, as does removing both blue nails, but removing one nail of each color leaves the picture hanging.

Puzzle 8 (1+2-out-of-2+2) Hang a picture on two red nails and two blue nails so that removing any one red nail fells the picture, as does removing both blue nails, but removing just one blue nail leaves the picture hanging.

Puzzle 9 (1+3-out-of-3+3) Hang a picture on three red nails and three blue nails so that removing any one red nail fells the picture, as does removing all three blue nails, but removing just one or two blue nails leaves the picture hanging.

Puzzle 10 (1+2-out-of-3+3) Hang a picture on three red nails and three blue nails so that removing any one red nail fells the picture, as does removing any two of the blue nails, but removing just one blue nail leaves the picture hanging.

Puzzle 11 (1+1-out-of-2+2+2) Hang a picture on two red nails, two green nails, and two blue nails so that removing two nails of different colors (one red and one green, or one red and one blue, or one green and one blue) fells the picture, but removing two nails of the same color leaves thepicture hanging.

My objective is to convey the information about nails for all 11 forms of the puzzle in 11 diagrams on kids level. I can use three colors (white, grey and black) only.

My attempt for puzzles 1-7 is

The solid and filled circle is a nail, the dotted circle is a removed nail at the first step, and the dotted and filled circle is a removed nail at the second step.

The R and G letters are nail's colors for the puzzle 7. The rectangle denotes the union of nails by color.

Edit 1 After proposed answers and comments I tried to change the idea of visualization.

The rectangle is a nail, the dotted line denotes a removed nail.

Question. Is the visualization of the puzzles clear or are there controversial issues?

Later I am going to visualize the all puzzles. But now I draw schemes for 1-8 puzzles.

• This seems really subjective. I'm not sure what exactly you're looking for here.
– Deusovi
Mar 10, 2020 at 0:52
• @deusovi, I am looking for an alternative idea, and possible errors.
– Nick
Mar 10, 2020 at 2:09
• @Rubio, I have updated the post.
– Nick
Mar 10, 2020 at 6:52

My concern with this whole approach is that you're trying to reduce to abstract figures some instructions that are fairly easy to explain textually but may prove somewhat more challenging to unambiguously depict visually, especially to children.

I would guess the objective is to reduce the instructions to this visual shorthand and then write/etch/whatever it onto the wood block you're creating for each puzzle, to depict which "nails" need to be removable to keep the rope captive vs. which one(s) should free it. But let's not forget that the image for Puzzle 2 really means

$$\raise 133pt\rlap{\textrm{not only}}$$ $$\raise 45pt{\textrm{but also}}$$

so that it may be confusing to see just the one picture but have to remember that it doesn't literally refer to those exact "nails" positionally.

Kids may struggle with that abstraction, and it would be a shame if what is actually a simple conceptual puzzle were to be made more difficult to understand by an attempt to, well, simplify the explanation.

Beyond that, the visual depictions for puzzles 7-10 and, especially, 11 seem like they're going to require increasingly confusing ways to represent the groupings allowed. The other answer here offers another way to visualize this but there again, it seems like we're trading one unintuitive abstraction for another. These are certainly going to have to be explained to a solver, particularly if that solver is a child.

All of which begs the question - if you're going to have to explain the puzzle anyway, why not just print the textual description on the puzzle instead? And if you just want some sort of short-hand reminder for what the puzzle is, why not use the ones the puzzles themselves already use?

Puzzle 1 (1-out-of-3)
Puzzle 2 (2-out-of-3)
Puzzle 3 (1+2-out-of-3)
Puzzle 4 (1-out-of-4)
Puzzle 5 (2-out-of-4)
Puzzle 6 (3-out-of-4)
Puzzle 7 (2+2-out-of-2+2)
Puzzle 8 (1+2-out-of-2+2)
Puzzle 9 (1+3-out-of-3+3)
Puzzle 10 (1+2-out-of-3+3)
Puzzle 11 (1+1-out-of-2+2+2)

I guess at the end of the day, it's just still not clear (to me anyway) what the actual objective here is, and why the more obvious seeming ways to describe these puzzles—the text ones you already listed—aren't good enough for what you're trying to accomplish.

• I agree with you that orders like (1, 2, 3), (2, 1, 3) and (1, 3, 2) do not have matter. I added the photo of puzzle and new idea of visualization.
– Nick
Mar 12, 2020 at 6:24

My idea for how to visualize this is to depict the nails as having interior regions which are partly filled and partly unfilled, with the conceit that whenever all of the removed nails, when overlapped with each other, together have a filled portion which fully covers the entire interior, the picture will drop.

I will demonstrate this for Puzzle 11:

As you can see from the picture, either red nail has a small, unfilled area in the lower right. If only the two red nails are removed, and their interiors are overlapped, the lower right unfilled area is still unfilled, and so the picture will not drop. However, if any one of the green or blue nails is then also removed, when the interiors are overlapped, that lower right area will now be filled, and so the picture will drop.

It is fairly easy to apply this method to most of the eleven scenarios you described, although I have not yet figured out a way to do it for Puzzle 6.

EDIT:

Here's a solution for Puzzle 6. It is not elegant at all, and it seems like there must be a simpler, more elegant way to do it, but this gets the job done for now. Choosing any two nails is not enough to fill all regions of the circle, but choosing any three will. Obviously, the way it is now would be quite difficult for the kids to deal with.

• I have the same concern with this approach that I do with OP's entire premise -- it seems to me that it'd take more effort and explanation to understand what these visualizations are trying to tell someone, a kid in particular since apparently that's the target audience, than it'd take to just put it in words.
– Rubio
Mar 10, 2020 at 7:50
• This method seems complicated at first, but once you work through a few examples and get the hang of it, it is not so difficult, I think. Kids would really take to it, I think, and it would be fun and stimulating for them. Mar 10, 2020 at 7:52
• @Rubio, I want to use the text description but also want to burn out pictures on the wood puzzle.
– Nick
Mar 10, 2020 at 7:54

I think the most straightforward pictorial description is to have a picture for each nail that is coloured segment of a disc. Any combination of nails for which the pictured segments can be put together to exactly form a disc of a single colour should release the string. So the puzzles would be:

Puzzle 1 (1-out-of-3) Three discs
Puzzle 2 (2-out-of-3) Three half-discs
Puzzle 3 (1+2-out-of-3) One disc and two half-discs
Puzzle 4 (1-out-of-4) Four discs
Puzzle 5 (2-out-of-4) Four half-discs
Puzzle 6 (3-out-of-4) Four 1/3-discs
Puzzle 7 (2+2-out-of-2+2) Two red half-discs and two blue half-discs
Puzzle 8 (1+2-out-of-2+2) Two red discs and two blue half-discs
Puzzle 9 (1+3-out-of-3+3) Three red discs and Three blue 1/3-discs
Puzzle 10 (1+2-out-of-3+3) Three red discs and Three blue 1/2-discs

Unfortunately puzzle 11 cannot be shown this way. You can however do it if the goal is now to create a multi-coloured disc.

Puzzle 11 (1+1-out-of-2+2+2) Two red half-discs, two green half-discs, and two blue half-discs.

So maybe the puzzles should also have a picture of the goal. For the first 10 puzzles it would just be a circle, but for 11 it would be a circle with a line through the middle to show the two halves need to be distinct.

Here is a picture:

• This was my original approach, but it didn't work for Puzzle 11, which is why I revised my method, wanting to find a single method that worked for all cases. But, yes, it is quite a simple method, so using it for only Puzzles 1-10, and a different method for Puzzle 11, is not a bad idea. Mar 10, 2020 at 9:38
• Referring to your revised post, which has the image included, the Puzzle 7 entry seems to be unclear, as it does not clearly denote that removing one of each color will NOT drop the picture. Mar 10, 2020 at 9:58
• Note OP said "I can use three colors (white, grey and black) only."
– Rubio
Mar 10, 2020 at 11:02
• @Rubio I overlooked that. Still, just using those three colours for the segments should be fine for puzzles 1-10. It is only for 11 that it would become confusing because it needs the goal configuration and that should not really be in any of the colours given with the nails. I suspect crosshatching or other decoration could be used to simulate other colours. Mar 10, 2020 at 11:08
• @JaapScherphuis, your disk's idea is very good, it gives me a rectangle representation.
– Nick
Mar 12, 2020 at 6:21